The test statistic \(t\) for the population Pearson correlation \(\rho\) (Greek letter rho) is estimated using the observed correlation \(r\).
\[
t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}
\]
Getting a p-value
Use the Student’s \(t\) distribution with degrees of freedom \(\text{df}=n-2\) to find a p-value for the observed correlation \(r\) in a sample of size \(n\) under the null hypothesis \(H_0 \colon \rho=0\).
# specify the test statistic and degrees of freedompt(test_statistic,df = n-2,lower.tail = F) # optional parameter
Eyeballing a Line
How do we find the best line to draw through variables that appear to have a linear relationship?
Quantifying Error: Residuals
Residuals are the difference between the observed values and the predicted values.
Special Topic: Correlation vs Causation
Research Question: Do ice cream sales cause sunburns?
Special Topic: Correlation vs Causation
As ice cream sales increase, the number of sunburns also increases
Strong, positive, linear correlation
High temperatures affect both ice cream consumption and behaviors that lead to sunburn
Confounding Variables
When you have a confounding variable, you might find dependence between two unrelated variables that are only connected by the confounder.
